Self-consistent System Of Equations Research Articles

MAXWELL’S UNIFICATION OF OPTICS AND ELECTROMAGNETISM AND THE INCOMMENSURABILITY TENET

It is contended that one of the promising directions for solving the problem of incommensurability of paradigms, coined by T. Kuhn and P. Feyerabend, can be associated with the trend of advancement of neo-Kantian epistemology, presented by the writings of Ernst Cassirer. According to Cassirer, the statements fixing connections and relationships between mathematical ideal constructs form a “neutral language” that can serve as a firm ground for comparing the “old” and “new” paradigms. A case study of the genesis and functioning of a neutral mathematical language related to the Maxwellian synthesis of optics and electromagnetism isprovided. It is elicited that its basis is constituted by the mathematical language of continuum mechanics. The main function of the neutral language was to project the consequences of all the unified theories onto the mathematical model, “rewrite” all known laws in this mathematical language, compare their conclusions with each other to eliminate contradictions and generalize the stuff in a self-consistent system of equations.

Starting currents of modes in cylindrical cavities with mode-converting corrugations for second-harmonic gyrotrons

A self-consistent system of equations (known as single-mode gyrotron equations) is extended to describe the beam-wave interaction in a cylindrical gyrotron cavity with mode-converting longitudinal corrugations, which produce coupling of azimuthal basis modes. The system of equations is applied to investigate the effect of corrugations on starting currents of the cavity modes. For these modes, eigenvalues, ohmic losses, field structure, and beam-wave coupling coefficients are investigated with respect to the corrugation parameters. It is shown that properly sized mode-converting corrugations are capable of improving the selectivity properties of cylindrical cavities for second-harmonic gyrotrons.

Development of the self-consistent approximation and its application to the problem of magnetoelastic resonance in an inhomogeneous medium

Our previously proposed approximation involving both the first and second terms of the expansion of the vertex function is generalized to the system of two interacting wavefields of different physical nature. A system of self-consistent equations for the matrix Green’s function and matrix vertex function is derived. On the basis of this matrix generalization of the new self-consistent approximation, a theory of magnetoelastic resonance is developed for a ferromagnetic model, where the magnetoelastic coupling parameter e(x) is inhomogeneous. Equations for magnetoelastic resonance are analyzed for one-dimensional inhomogeneities of the coupling parameter. The diagonal and off-diagonal elements of the matrix Green’s function of the system of coupled spin and elastic waves are calculated with the change in the ratio between the average value e and rms fluctuation Δe of the coupling parameter between waves from the homogeneous case (e ≠ 0, Δe = 0) to the extremely randomized case (e = 0, Δe ≠ 0) at various correlation wavenumbers of inhomogeneities k c. For the limiting case of infinite correlation radius (k c = 0), in addition to approximate expressions, exact analytical expressions corresponding to the summation of all diagrams of elements of the matrix Green’s function are obtained. The results calculated for an arbitrary k c value in the new self-consistent approximation are compared to the results obtained in the standard self-consistent approximation, where only the first term of the expansion of the vertex function is taken into account. It is shown that the new approximation corrects disadvantages of the Green’s functions calculated in the standard approximation such as the dome shape of resonances and bends on the sides of resonance peaks. The appearance of a fine structure of the spectrum in the form of a narrow resonance on the Green’s function of spin waves and a narrow antiresonance on the Green’s function of elastic waves, which was previously predicted in the standard self-consistent approximation, is confirmed. With an increase in the parameter k c, the Green’s functions calculated in the standard and new approximations approach each other and almost coincide with each other at k c/k ≥ 0.5. At the same time, the results of this work indicate that the new self-consistent approximation has a certain advantage for studying the problems of stochastic radiophysics in media with long-wavelength inhomogeneities (small k c values), because it describes both the shape and width of peaks much better than the standard approximation.

Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli.

Information processing in the brain crucially depends on the topology of the neuronal connections. We investigate how the topology influences the response of a population of leaky integrate-and-fire neurons to a stimulus. We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network. We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing. We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

Thermodynamics of Ising rare-earth magnet in the static fluctuation approximation

The authors obtain the closed self-consistent system of equations for calculation of the desired thermodynamic values of the Ising magnet with arbitrary spin and in presence of the single-ion anisotropy in the frame of the original static fluctuation approach (SFA). The essence of the SFA is based on accurate calculations of fluctuations of a molecular field. As an example the complete analysis of the critical behavior of the Ising magnet with spin S = 1 (Blume-Capiel model) is shown. The expressions for the phase transitions lines of the first and the second order are found. In addition, the coordinates of the tricritical point for some types of cubic lattices are found also.

Electromagnetic waves in an axion-active relativistic plasma non-minimally coupled to gravity

We consider cosmological applications of a new self-consistent system of equations, accounting for a nonminimal coupling of the gravitational, electromagnetic and pseudoscalar (axion) fields in a relativistic plasma. We focus on dispersion relations for electromagnetic perturbations in an initially isotropic ultrarelativistic plasma coupled to the gravitational and axion fields in the framework of isotropic homogeneous cosmological model of the de Sitter type. We classify the longitudinal and transversal electromagnetic modes in an axionically active plasma and distinguish between waves (damping, instable or running), and nonharmonic perturbations (damping or instable). We show that for the special choice of the guiding model parameters the transversal electromagnetic waves in the axionically active plasma, nonminimally coupled to gravity, can propagate with the phase velocity less than speed of light in vacuum, thus displaying a possibility for a new type of resonant particle-wave interactions.

Strong-coupling theory for multiband superconductors

We consider a microscopic theory of the strong coupling in multiband superconductors with an arbitrary electron-boson interaction. Based on the method of the equations of motion for two-time Green’s functions, we derive the Dyson equation with the self-energy operator in the form of the multiparticle Green’s function taking the interaction of electrons with phonons and spin fluctuations into account. We obtain a self-consistent system of equations for the normal and anomalous components of the Green’s function and the self-energy operator calculated in the approximation of noncrossing diagrams. We discuss the approximate solution of the system of equations taking only components of the self-energy operator that are diagonal with respect to the band index into account for studying superconductivity in iron-based compounds.

Magnetic states of single impurity in disordered environment

The charged and magnetic states of isolated impurities dissolved in amorphous metallic alloy are investigated. The Hamiltonian of the system under study is the generalization of Anderson impurity model. Namely, the processes of elastic and non-elastic scattering of conductive electrons on the ions of a metal and on a charged impurity are included. The configuration averaged one-particle Green's functions are obtained within Hartree-Fock approximation. A system of self-consistent equations is given for calculation of an electronic spectrum, the charged and the spin-polarized impurity states. Qualitative analysis of the effect of the metallic host structural disorder on the observed values is performed. Additional shift and broadening of virtual impurity level is caused by a structural disorder of impurity environment.

Microwave excitation of submerged plasma bubbles

Steady-state models were formulated for spherical plasma bubbles, excited in water by an external microwave radiation source. The bubbles were assumed to be at rest relative to the surrounding water, and the energy absorbed by the plasma was balanced by energy loss through thermal conductance and convection to the surrounding water. Two regimes were considered: (1) bubbles with radii R less than the skin depth δ, i.e. R < δ, and (2) bubbles with R > δ. A self-consistent system of equations was formulated. In the R < δ case, a considerable difference was found between electron temperature and the temperature of the heavy plasma components, but this difference decreased with increasing radius. The calculations showed that the bubble radius decreased with increasing electric field amplitude, but R in the R > δ case obtained for strong electric fields was larger than in the R < δ weak field case. The R > δ bubbles had an isothermal core and power from the microwave field was absorbed in a relatively thin layer in the plasma bubble.

Theory of the residual electrical resistivity of binary actinide alloys

A system of self-consistent equations has been proposed for the coherent potential approximation of the multiband conductivity model for the case of conduction electron scattering from chaotic electric fields of ions of disordered binary alloy components at zero temperature. It has been qualitatively demonstrated that the deviation of the concentration dependence of the residual electrical resistivity of actinide alloys with multiband conductivity from the Nordheim rule is caused by the explicit dependence of the electrical resistivity of the alloy on the magnitude and sign of the real part of the Green’s function at the Fermi level. The derived system of equations for the multiband coherent potential approximation has been used to calculate the concentration dependence of the density of states and the residual electrical resistivity of the alloys of neptunium and plutonium. The results of the calculations have been compared with the available experimental data.

Temperature and concentration dependences of the electrical resistivity for alloys of plutonium with americium under normal conditions

The temperature and concentration dependences of the electrical resistivity for alloys of americium with plutonium are analyzed in terms of the multiband conductivity model for binary disordered substitution-type alloys. For the case of high temperatures (T > ΘD, ΘD is the Debye temperature), a system of self-consistent equations of the coherent potential approximation has been derived for the scattering of conduction electrons by impurities and phonons without any constraints on the interaction intensity. The definitions of the shift and broadening operator for a single-electron level are used to show qualitatively and quantitatively that the pattern of the temperature dependence of the electrical resistivity for alloys is determined by the balance between the coherent and incoherent contributions to the electron-phonon scattering and that the interference conduction electron scattering mechanism can be the main cause of the negative temperature coefficient of resistivity observed in some alloys involving actinides. It is shown that the great values of the observed resistivity may be attributable to interband transitions of charge carriers and renormalization of their effective mass through strong s-d band hybridization. The concentration and temperature dependences of the resistivity for alloys of plutonium and americium calculated in terms of the derived conductivity model are compared with the available experimental data.

Storage of intense optical pulses in solids

Possibility to store optical information in media with fast relaxations is studied in case of arbitrary intensities of probe and control pulses. Analytical solution of the self-consistent system of equations for propagation and density matrix is obtained with allowance for the first non-stationary corrections. A new scheme of storage is proposed which allows an essential reduction of the optical length needed for storage.

Studying the properties of carbon nanotubes with the functional integration method

We obtain an exact, closed, self-consistent system of equations for describing nanotubes that takes electron and oscillation subsystems in the collective variables into account. Collective excitations in nanotubes are described by the quantum numbers n, m, and k, where n − 1 is the number of radial modes, m is the number of azimuthal modes, and k − 1 is the number of longitudinal modes of the wave function. The results obtained approximate the experimental data better than those obtained by the method of linear combinations of atomic orbitals.

Domain structure of a planar layer of a nematic liquid crystal in oscillating Couette flow

The spatially periodic structure arising in a planar layer of a nematic liquid crystal in oscillating Couette flow is described theoretically. The analysis of this effect is based on the hydrodynamic equations for nematic liquid crystals, from which a self-consistent system of equations is singled out for perturbations of hydrodynamic variables. It is shown that the structure and threshold parameters of the effect depend on the oscillation frequency and the layer thickness through the scaling combination ωh2. The dependence of the configuration of distortions on the viscosity α3 is analyzed.

Nonlinear acoustic transparency phenomena in strained paramagnetic crystals

Nonlinear dynamics of longitudinal acoustic pulses propagating in a strained paramagnetic cubic crystal at low temperature is analyzed. The direction of constant uniform strain is parallel to one of the fourth-order symmetry axes. Effective-spin 1 ions are considered as paramagnetic impurities with the strongest spin-lattice interaction. In this medium, normally degenerate magnetic sublevels are dynamically shifted by the quadrupole Stark effect, and the frequencies of the transitions induced by an acoustic pulse change accordingly. The self-consistent system of equations derived in this study without using the slowly varying envelope approximation describes pulse propagation at an arbitrary angle to the direction of the static strain. Exponentially and rationally decaying monopolar and breather-like solutions to the system are obtained. An analysis of the solutions reveals an asymmetry of the pulse polarity depending on the type of strain (extension or compression) and the pulse propagation direction. In particular, it is shown that acoustic transparency associated with monopolar strain pulses exhibits threshold behavior. The sign of the time area (zeroth harmonic) of breather-like strain pulses is such that the transition frequency averaged over an oscillation period dynamically decreases. This behavior determines the efficiency of generation of high-order harmonics of acoustic pulses in strained paramagnetic crystals.

Cyclotron suppression of generation in a relativistic backward-wave oscillator of Čerenkov type

We consider the self-consistent system of equations describing the combined interaction of an eigenwave of the periodic structure with the space-charge waves and one of the cyclotron waves of a magnetized relativistic electron beam. A planar model of a relativistic backward-wave oscillator is studied. The main attention is paid to quenching of self-oscillations due to excitation of fast cyclotron waves. The appearance of hard oscillation modes and hysteresis phenomena in the backward-wave oscillator is observed when the value of the guiding magnetic field is close to the resonant one.

Domain structure of a nematic-liquid-crystal layer in low-frequency Couette flow

The spatially periodic structure arising in a homeotropic nematic-liquid-crystal (NLC) layer in low-frequency Couette flow is described theoretically. The analysis of this phenomenon is based on the hydrodynamic equations for NLCs, from which a self-consistent system of equations is selected for perturbations of hydrodynamic variables: the steady-state angle of the molecule rotation, the liquid flow, and the velocity of oscillating vortex flows. The formation of the periodic structure is explained by the phase delay of the velocities of the vortex oscillating flows forming in the deformed structure with respect to the shear velocity in the Couette flow. It is shown that at low frequencies this difference in the velocities is caused by the orientational waves near the layer boundaries. In the case of fixed orientation of molecules at the boundaries, the dependence of the threshold shear amplitude on the frequency and layer thickness is given by the relation uth ∼ (ωh2)−1/4. The influence of the conditions for the molecule orientation at the layer boundaries on the above phenomenon is analyzed.

Self-action of laser radiation under the conditions of electromagnetically induced transparency

The laser-radiation self-action dynamics in the electromagnetically induced transparency band were studied analytically and numerically for an atomic gas with the Λ-type energy-level scheme. The self-consistent system of equations describing the spatiotemporal evolution of a finite-amplitude wave packet in the field of a uniform pump wave was derived. The self-action in the presence of competing diffraction, nonlinear dispersion, and radiation absorption in the system was qualitatively analyzed; in particular, the conditions for self-focusing of a probe beam were determined. The results were confirmed by the numerical simulation of wave-packet spatiotemporal evolution.

Dynamical Spin Susceptibility in the t – J Model in the Superconducting Phase

Dynamical spin susceptibility is calculated for the t–J model in the superconducting phase using the memory function method in terms of the Hubbard operators. The self-consistent system of equations for the memory function is obtained within the mode-coupling approximation. Both itinerant hole excitations and localized spin fluctuations contribute to the memory function. Moreover, the itinerant contribution itself consists of two parts, i.e., the contribution of Bogoliubov quasiparticles and that of Cooper pairs. The spin dynamics is diffusive in the hydrodynamic limit, but the itinerant part does not contribute to the spin diffusion. In the high frequency region, spin–wave-like excitations continue to exist. We discuss our analytic results in the light of neutron scattering experiments performed on the cuprate superconductors.

Self-consistent field model for spatially inhomogeneous Bose systems

A self-consistent field (SCF) model is constructed for spatially inhomogeneous Bose systems with broken symmetry. A system of self-consistent equations is obtained for the wave functions of the quasiparticles and the wave function of the condensate particles and another system of equations for the normal and anomalous single-particle density matrices. The many-particle wave function is found. The thermodynamics of multiparticle Bose systems is constructed on the basis of a microscopic treatment in the SCF approximation. It is stressed that Bose condensation in the ideal gas model is substantially different from Bose condensation in a system of interacting Bose particles because a pair condensate is necessarily present as well as the single-particle condensate, even for an arbitrarily weak interaction. The role of single-particle and collective excitations in Bose systems is discussed.